Narrow Results

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the graph.

We classify Cohen–Macaulay graphs of girth at least 5 and planar Gorenstein graphs of girth at least 4. Moreover, such graphs are also vertex decomposable.

For a simplicial complex Δ, we introduce a simplicial complex attached to Δ, called the expansion of Δ, which is a natural generalization of the notion of expansion in graph theory. We are interested in knowing how the properties of a simplicial complex and its Stanley–Reisner ring relate to those of its expansions. It is shown that taking expansion preserves vertex decomposable and shellable properties and in some cases Cohen–Macaulayness. Also it is proved that some homological invariants of Stanley–Reisner ring of a simplicial complex relate to those invariants in the Stanley–Reisner ring of its expansions.

For every simple graph $G$, a class of multiple clique cluster-whiskered graphs $G^{e_{\pi }\mathfrak{m}}$ is introduced, and it is shown that all such graphs are vertex decomposable; thus, the independence simplicial complex $\mathrm{Ind}{G}^{e_{\pi }\mathfrak{m}}$ is sequentially Cohen–Macaulay. The properties of the graphs $G^{e_{\pi }\mathfrak{m}}$ and $G^{\pi }$ constructed by Cook and Nagel are studied, including the enumeration of facets of the complex $\mathrm{Ind}{G}^{\pi }$ and the calculation of Betti numbers of the cover ideal $I_c(G^{e_{\pi }\mathfrak{m}})$. We also prove that the complex$\mathrm{\Delta }=\mathrm{Ind}H$ is strongly shellable and pure for either a Boolean graph $H=B_n$ or the full clique-whiskered graph $H=G^W$ of $G$, which is obtained by adding a whisker to each vertex of $G$. This implies that both the facet ideal $I(\mathrm{\Delta })$ and the cover ideal $I_c(H)$ have linear quotients.